To detect objects in a region of interest (ROI), radar antennas transmit pulses to illuminate the ROI, and receive reflected echoes for an imaging process. The received echoes can be approximated as a weighted sum of delayed versions of the transmitted pulses, with weights related to the reflectivities of the objects and delays related to the ranges of the objects relative to the antennas. Radar imaging is basically an inverse problem to solve for the weights and the delays given the transmitted pulses and the received echoes. When the locations of transmit and receive antennas are known, a simple delay-and-sum method can generate a well-resolved image of the ROI with a sufficient radar aperture size.
However, in radar applications, it is very common that the antenna locations are not known accurately due to environment interference or imprecise motion control of the radar platform. Although modern navigation systems such as Global Positioning System (GPS) can measure positions with high accuracy, the possible position errors are still beyond the requirement of high-resolution radar imaging.
For example, for vehicle mounted mono-static radar systems, as the vehicle is moving along some predesigned trajectory, position perturbations can be introduced due to non-smooth road surface or varying driving velocity. These position perturbations can be as large as several times the wavelength of the radar center frequency. In such situation, the virtual radar array is no longer uniform and the position errors need to be compensated in the imaging process. Otherwise, the objects to be detected are not focused, or even unresolvable when the position perturbations are greater than the wavelength of the central frequency. Therefore, it is desirable to perform autofocus imaging to achieve a well focused radar image especially when the antenna perturbations are relatively large.
Autofocus (AF) is a challenging problem in radar imaging as well as other array imaging applications using different sensor modalities. The existing AF methods can be roughly grouped into two categories. One is based on phase compensation, the other is based on position or motion compensation.
Phase-compensation based AF methods compensate data phases in terms of different merits, such as minimum entropy or least squares to generate a well-focused image. Phase-compensation based methods generally work well in compensating environment-induced phase distortion. However, for antenna position-induced phase error, which changes from object to object, simple phase-compensation cannot generate well focused image. In particular, when the area size of imaging domain increases, phase compensation methods can focus well at a particular area, but de-focus at other areas. Motion compensation based methods, on the other hand, seek to compensate for the position such that different position-induced phase errors can be corrected. However, it is difficult to achieve a global optimal solution in estimating antenna positions for AF imaging.
Compressive sensing (CS) based AF methods can concurrently perform AF imaging and compensate position errors by imposing sparsity of the image to be reconstructed. Because the position error is unknown, CS-based AF methods model the imaging problem as an optimization problem with a perturbed projection matrix. The corresponding optimal solution, however, is with error bound related to the position error. A global optimal solution is only achievable when the position error is much smaller than the wavelength, and with a good initialization. When the position errors are in the order of several wavelengths, those methods cannot converge to a focused image.